For what values of x is #f(x)=-2x- (x+3)^ (-2/3)# concave or convex?

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Answer 1 · 2017-06-18T19:38:47

#x in (-infty,-3)#: #f(x)# is concave down

#x in (-3,+infty)#: #f(x)# is concave down

Find the first derivative, which is the slope of the curve

#f'(x)=-2+2/3(x+3)^(-5/3)#

The second derivative, is the rate of change of the curve

#f''(x)=-10/9(x+3)^(-8/3)=(-10)/(9root(3)((x+3)^8))#

Whenever #f''(x) < 0#, the curve is concave downward (like a frown face) and whenever #f''(x) > 0#, the curve is concave upward (like a smiley face).

Testing points around #x=-3# (where #f''(x)# divides by zero).

#x < -3#: All points are negative; Concave down

#x > -3#: All points are negative; Concave down